Geometry Question Points on a Parabola

Question

We have the curve with equation $y^2=x+4$. Point $A$ is on the curve and has coordinates $(0,2)$, $B$ and $C$ are other points lying on the curve where the line segment $AB$ is perpendicular to $BC$. What values can the $y$-coordinate of the point $C$ be?

Answer 1

Let $B(b^2-4,b)$ and $C(c^2-4,c)$, thus:

$$m_{AB}=\frac{2-b}{4-b^2}=\frac{1}{2-b}$$

$$m_{BC}=\frac{b-c}{b^2-c^2}=\frac{1}{b-c}$$

Perpendicularity condition:

$$m_{AB}=-\frac{1}{m_{BC}}\implies c-b=\frac{1}{2-b}\implies c=b+\frac{1}{2-b}$$